# algebraic manifold

###### Definition.

Let $k$ be a field and let $M\subset k^{n}$ be a submanifold. $M$ is said to be an algebraic manifold (or $k$-algebraic) if there exists an irreducible algebraic variety $V\subset k^{n}$ such that $\dim V=\dim M$ and $M\subset V$. If $k=\mathbb{R}$, then $M$ is called a Nash manifold.

It can be proved that such a manifold is defined as the zero set of a finite collection of analytic algebraic functions.

## References

• 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
Title algebraic manifold AlgebraicManifold 2013-03-22 15:36:08 2013-03-22 15:36:08 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 14P20 msc 14-00 msc 58A07 algebraic submanifold $k$-algebraic manifold $k$-algebraic submanifold Nash manifold Nash submanifold